### Summary

- Top of page
- Summary
- Introduction
- Asymptotic analysis
- The limits of asymptotic analysis
- Transient analysis
- Models as part of the applied ecologist’s toolbox
- Conclusion
- Acknowledgements
- References
- Supporting Information

**1.** Matrix population models are tools for elucidating the association between demographic processes and population dynamics. A large amount of useful theory pivots on the assumption of equilibrium dynamics. The preceding transient is, however, of genuine conservation concern as it encompasses the short-term impact of natural or anthropogenic disturbance on the population.

**2.** We review recent theoretical advances in deterministic transient analysis of matrix projection models, considering how disturbance can alter population dynamics by provoking a new population trajectory.

**3.** We illustrate these impacts using plant and vertebrate systems across contiguous and fragmented landscapes.

**4.** Short-term responses are of fundamental relevance for applied ecology, because the time-scale of transient effects is often similar to the length of many conservation projects. Investigation of the immediate, post-disturbance phase is vital for understanding how population processes respond to widespread disturbance in the short- and into the long term.

**5.** *Synthesis and applications*. Transient analysis is critical for understanding and predicting the consequences of management activities. By considering short-term population responses to perturbations, especially in long-lived species, managers can develop more informed strategies for species harvesting or controlling of invasive species.

### Introduction

- Top of page
- Summary
- Introduction
- Asymptotic analysis
- The limits of asymptotic analysis
- Transient analysis
- Models as part of the applied ecologist’s toolbox
- Conclusion
- Acknowledgements
- References
- Supporting Information

Matrix projection analysis is a flexible tool for incorporating the life history of an organism into a structured population model (Caswell 2001). It has been influential in elucidating how demographic processes impact population dynamics in evolutionary ecology (Lande 1982; van Tienderen 2000) and conservation biology (Morris & Doak 2002). The modelled asymptotic growth λ_{1} is an especially important population parameter in matrix projection analysis. In a constant environment, λ_{1} > 1 indicates that the population will eventually increase whereas λ_{1} < 1 indicates that the population will decline to extinction. The assumption of fixed demographic rates of survival, fertility and dispersal fails in natural populations, provoking development of stochastic alternatives (Tuljapurkar 1990; Lande, Engen & Sæther 2003). However, it is not always necessary for the conservation biologist to use complex and data-intensive stochastic models: no model – no matter how complex – can hope to mirror biological phenomena exactly (Levins 1966). Simple models offer conceptual clarity, and are an increasingly influential tool for rapid diagnosis of the persistence probability of threatened populations and species (Milner-Gulland & Rowcliffe 2007) or managing the spread of re-introduced or non-native species (Bullock, Pywell & Coulson-Phillips 2008). Matrix-modelling approaches that address questions like these often assess consider only equilibrium properties, but populations of conservation concern may rarely be in an equilibrium state (Hastings 2004).

We review theoretical advances that focus specifically on the short-term (i.e. transient), rather than equilibrium (i.e. asymptotic), state of the system. The deterministic approaches we consider can provide substantial insight, and have the attraction of moving beyond simplistic equilibrium approaches whilst maintaining the analytical tractability often lost in stochastic modelling. We outline the foundations of asymptotic analysis of population projection matrices (Caswell 2001 describes model construction extensively), and then describe and illustrate the relevance of transient population dynamics in applied ecology.

### Asymptotic analysis

- Top of page
- Summary
- Introduction
- Asymptotic analysis
- The limits of asymptotic analysis
- Transient analysis
- Models as part of the applied ecologist’s toolbox
- Conclusion
- Acknowledgements
- References
- Supporting Information

Despite criticisms concerning its lack of biological reality (e.g. Stephens *et al.* 2002), asymptotic analysis of matrix population models produces a number of informative quantities for applied ecology (Table 1). Asymptotic growth λ_{1} can provide an accurate assessment of the best way to control an invasive species (Bullock, Clear Hill & Silvertown 1994), the effect of different herbicide treatments (Crone, Marler & Pearson 2009), whether harvesting of a species is sustainable (Ghimire *et al.* 2008) or if a particular re-introduction approach is the most effective for a declining species (Linares, Coma & Zabala 2008). Such comparisons can be taken further using sensitivities or elasticities (Table 1). Retrospective perturbation analysis, such as a Life Table Response Experiment, suggests how some desired difference among populations can be managed for (Bruna & Oli 2005) or why certain populations are expanding or declining to extinction (Cooch, Rockwell & Brault 2001; Nicole, Brzosko & Till-Bottraud 2005). A related, but philosophically different approach (Caswell 2000), is prospective perturbation analysis. Prospective analysis does not indicate how the underlying demographic rates respond directly to the environment, revealing simply what the effect on population growth would be if they were changed (Silvertown, Franco & Menges 1996). Prospective analysis can also be used to inform management, for example in determining the least damaging harvesting strategy for an exploited species (Rogers-Bennett & Leaf 2006), how to enhance a particular life stage transition (Norris & McCulloch 2003) or how best to target control of an invasive species (Shea & Kelly 1988).

Table 1. Summary of the asymptotic properties of matrix models (see Caswell 2001 for full details) Measure | Symbol | Definition | Biological meaning |
---|

Asymptotic growth | λ_{1}, or *r* = ln(λ_{1}) | The dominant (largest) eigenvalue of the population transition matrix *A* | Eventual population growth rate |

Stable distribution | *w* | The right eigenvector associated with λ_{1}, rescaled such that all elements sum to unity | The post-transient proportion in each class (age or stage) |

Reproductive value | *v* | The left eigenvector associated with λ_{1}, rescaled such that all elements are relative to the first | Mean number of offspring produced from a post-transient individual in each class |

Sensitivity of λ_{1} to a matrix element | | Linear approximation of the association between a matrix element *a*_{ij} and λ_{1}. Assumes linearity, hence infinitesimal perturbation and stable population structure. Lower-level sensitivities to the demographic rates constitute matrix elements can be calculated | The influence of *a*_{ij} on λ_{1} |

Elasticity of λ_{1} to a matrix elements | | As above, except association on a relative scale | The *relative* influence *a*_{ij} on λ_{1}, enabling direct comparisons of demographic rates between survival (bounded between 0 and 1) and fecundities (bounded below at 0 only) |

### The limits of asymptotic analysis

- Top of page
- Summary
- Introduction
- Asymptotic analysis
- The limits of asymptotic analysis
- Transient analysis
- Models as part of the applied ecologist’s toolbox
- Conclusion
- Acknowledgements
- References
- Supporting Information

Two key assumptions of asymptotic analyses are: (i) linear perturbations in sensitivity or elasticity analysis; and (ii) a stable population structure (Table 1). Any change to a vital rate will have nonlinear consequences for population growth (Hodgson & Townley 2004) and the assumption of linear perturbations infers infinitesimal change, making extrapolations to large increases (say 20%) problematic. Hodgson & Townley’s (2004) transfer-function approach assesses the inaccuracy of analysis based on linear perturbations and also of demographic rate independence. (“Integrated sensitivities”, [van Tienderen 1995] also correct for dependence among demographic rates, but assume linear perturbations). Although applying nonlinear perturbations can affect model prediction (Carslake, Townley & Hodgson 2009), neglecting a dynamic population structure has been argued to alter conclusions more (Caswell 2001, p. 615).

If the age-specific survival and fecundity rates are constant, the discrepancy between observed and asymptotic population structures declines exponentially over time as the population converges on the stable age distribution. The initial conditions of two populations might be very different, yet if the demographic rates remain the same both will converge on identical population structures (Cohen 1979b). However, population projections for the two populations might change markedly due to transient factors before convergence: there will be different proportions of individuals in each part of the population during this transient than under asymptotic, equilibrium conditions. Asymptotic population structure infers constant cohort size, but populations whose dynamics are influenced by strong cohort effects are widespread (Lindström & Kokko 2002) and the growth rates of such populations can change markedly from one cohort to the next (Gaillard *et al.* 1997). To highlight the effect that consideration of a non-equilibrium population structure has on the influence of demographic rates on λ_{1}, Coulson *et al.* (2004) compared elasticities that were weighted by the asymptotic, equilibrium population structure (i.e. the stable age distribution) with elasticities calculated using the observed population structure in a given year. Simply changing one part of the calculation enables the effect of the assumption of equilibrium structure on demographic inference to be determined.

To illustrate how interpretation can change when dynamic structures are considered, we constructed post-breeding population transition matrices using individual-based data collected since 1985 from the population of Soay sheep (*Ovis aries*) living in Village Bay on Hirta in the St. Kilda archipelago, Scotland (57º49′, 8º34′; see Clutton-Brock & Pemberton (2004) for comprehensive information on data-collection protocols). Each population transition matrix **A** takes the form given in Ezard *et al.* (2008) and a different **A** was constructed for each year under consideration using observed demographic rates from that year. Population size is defined here as the number of sheep alive on 1 August annually, which is the boundary between each ‘sheep year’. We calculated elasticities using the asymptotic and observed population structures for each year and refer to these quantities as equilibrium and non-equilibrium elasticities, respectively (Coulson *et al.* 2004).

The use of asymptotic elasticities does not optimally describe population processes in this case because the population does not appear to be converging on a stable equilibrium due to fluctuations in abundance and age structure (Clutton-Brock & Coulson 2002). The correlation between equilibrium and non-equilibrium elasticities of λ_{1} was significant but the variance explained low (Fig. 1; β = 0·401, SE = 0·073, *P *<* *0·001, *r*^{2} = 0·158, from a GLM with identity link and a squared variance function, the latter selected because the variance around non-equilibrium elasticities increased nonlinearly as elasticities increased). In particular, the contributions of lambs to λ_{1} are consistently overestimated when elasticities are calculated using the asymptotic rather than observed population structure, whereas the converse is true for prime-aged individuals, whose relative influence on long-term population growth is consistent across years and population structures (Ezard *et al.* 2008). Thus, in this example, the use of asymptotic elasticities might direct more management and conservation efforts towards lambs compared with efforts driven by an analysis that incorporates a dynamic population structure. This sort of insight is arguably highly relevant to the management of ungulates in eco-tourism and hunting industries.

### Models as part of the applied ecologist’s toolbox

- Top of page
- Summary
- Introduction
- Asymptotic analysis
- The limits of asymptotic analysis
- Transient analysis
- Models as part of the applied ecologist’s toolbox
- Conclusion
- Acknowledgements
- References
- Supporting Information

Management recommendations from mathematical models and achievable field protocols are rarely concordant. The power of models often lies in their ability to determine the population consequences, as assumed by a model, of management actions. Rather than a start- or end-point, population modelling ideally forms part of an integrated approach to applied ecology (Milner-Gulland & Rowcliffe 2007). Appropriate perturbation analysis might identify a key demographic rate, which in turn might become the focus of data collection efforts, which in turn might identify model limitations or flawed assumptions. Exceedingly complex models (e.g. stochastic, spatially-explicit, individual-based models) are often used in conservation, but may not always be necessary: simpler models can often provide sufficient detail and accuracy (Bullock *et al.* 2008). A rebuttal to the critique of a lack of biological realism in asymptotic analysis (e.g. Stephens *et al.* 2002) often lies in the flexibility of matrix modelling, which can eschew asymptotic analysis whilst retaining the analytical tractability of deterministic approaches. Examples include incorporation of interactions between harvesting and density dependence (Barbraud *et al.* 2008), functional relationships between predator demographic rates and prey abundance (Henden *et al.* 2008) or achievable change per unit cost (Baxter *et al.* 2006).

Transient analysis of matrix models can be used to estimate relevant elasticities amidst density dependence, environmental stochasticity and many other conservation-relevant scenarios (Caswell 2007). Analysis of transient dynamics forms a crucial part of the applied ecologist’s toolbox, potentially elucidating why asymptotic analysis can fail to approximate satisfactorily the short-term, on-the-ground reality of conservation and management. The differences between transient and asymptotic analysis reduce as the short-term blends into the medium- and long term (Cohen 1979a,b). The boundaries between these time frames depend, in part, upon the life history of the organism. Theoretical studies suggest that organisms with long generation times are more apt to experience: (i) longer durations of transient dynamics (Koons *et al.* 2005); (ii) larger departures in transient abundance and growth rates away from asymptotic conditions (Koons *et al.* 2005; Haridas & Tuljapurkar 2007); and (iii) larger magnitudes of population momentum relative to those with short generation times (Koons, Grand & Arnold 2006a; Koons *et al.* 2006b).

When considering mathematical models, ‘it is important to match the time-scale of observation with the analysis’ (Hastings 2004, p. 40). In applied ecology, the period described by the transient dynamics is likely to be the fundamental period of interest because it considers explicitly the ramifications of disturbance originating from abiotic, biotic and/or anthropogenic processes. Despite this, transients are not the be-all and end-all: both asymptotic and transient analyses assume a single perturbation to the system over the timeframe under consideration. This may be no more likely over 5 years than 50 or 500. Stochastic transient analysis has a role to play in systems characterized by numerous and repetitive disturbance, but may be less than informative in non-data rich systems when reliant on uninformed assumptions.